Poincaré recurrence theorem

Let (X,𝒮,μ) be a probability spaceMathworldPlanetmath and let f:XX be a measure preserving transformation.

Theorem 1.

For any ES, the set of those points x of E such that fn(x)E for all n>0 has zero measure. That is, almost every point of E returns to E. In fact, almost every point returns infinitely often; i.e.

μ({xE: there exists N such that fn(x)E for all n>N})=0.

The following is a topological version of this theorem:

Theorem 2.

If X is a second countable Hausdorff space and S contains the Borel sigma-algebra, then the set of recurrent points of f has full measure. That is, almost every point is recurrent.

Title Poincaré recurrence theorem
Canonical name PoincareRecurrenceTheorem
Date of creation 2013-03-22 14:29:50
Last modified on 2013-03-22 14:29:50
Owner Koro (127)
Last modified by Koro (127)
Numerical id 6
Author Koro (127)
Entry type Theorem
Classification msc 37B20
Classification msc 37A05